This is of course a restatement of the Banach-Tarski theorem, which tells us that a solid ball can be partitioned into finitely many non-overlapping subsets, which can then be rearranged to form two balls each of the same size as the original ball.
Of course, you couldn't do this with actual bananas (even though a topologist can't tell a banana from an orange, or a doughnut from a coffee cup, or eir pants from a hole in the ground). Damn atoms.
What other fruit-related math jokes are there? The only one I can think of is "What's purple and commutes? An abelian grape," which barely qualifies as a joke.
I found the Bananach-Tarski paradox via the Facebook group I support the right to choose one element from each set in a collection, which asks:
Do you believe that an infinite product of nonempty sets should be nonempty? Do you feel that non-measurable subsets of the reals should exist? Or games of perfect information with no winning strategy for either player? Do you believe that every set should have a well-ordering, or that any poset in which every chain has an upper bound is entitled to a maximal element? Do nonzero rings have the right to a maximal ideal? Is every vector space entitled to a basis? Should fields have algebraic closures? Should products of compact topological spaces be compact, and countable unions of countable sets be countable? Do you want to be able to cut a sphere up into a finite number of pieces and reassemble them, with only rigid motions, into a sphere twice as large?
Finally, because this blog is at least nominally about probability: imagine a (randomized) algorithm that outputs a string of letters as follows: start with any three letters a1, a2, a3. Then for each n > 3, consider some corpus of English text, and consider all the occurrences of the string an-3 an-2 an-1 in that corpus. Tabulate the distribution of the letter that follows that string in the corpus. Choose an from that distribution.
I don't have the programming skills or the corpus to work from, but I bet there's a significant chance that if you start with BAN you get BANANANANANA...